Problem: What is the inverse of the function $f(x)=3x-2$ ? $f^{-1}(x)=$
Answer: Let's start by replacing $f(x)$ with $y$. $y=3x-2$ If a function contains the point $(a,b)$, the inverse of that function contains the point $(b,a)$. So if we swap the position of $x$ and $y$ in the equation, we get the inverse relationship. In this case, the function is $y=3x-2$, so the inverse relationship is $x=3y-2$. Solving this equation for $y$ will give us an expression for $f^{-1}(x)$. $\begin{aligned} x&=3y-2\\\\ x+2&=3y\\\\ \dfrac{x+2}{3}&=y\\\\\\ \end{aligned}$ The inverse of the function is $f^{-1}(x)=\dfrac{x+2}{3}$. [I saw someone solve this problem by originally solving for x. Were they wrong?]